4.9 Application: Hückel Theory
87
Fig. 4.7 Hückel orbital
energy spectrum of benzene
as a function of index k,with
allowed values 0, ±1, ±2, 3
=
1
N
N −1
j =0
α + β
exp(−2πik/N)+ exp(+2πik/N)
= α + 2β cos(2πk/N)
(4.119)
The energies are thus seen to form N discrete levels, which are points on a cosine
curve, as shown in Fig. 4.7. Except for k = 0, and in the case of N even, k = N/2,
all levels E k and E −k are twofold-degenerate. Closed-shell structures thus will be
realized for N = 4n + 2, which is the famous Hückel condition for aromaticity.
These cyclic labels can easily be expanded to the full irrep designations of the D 6h
symmetry group for benzene. The atomic p z -orbitals transform as b 1 in the C 2v site
group. In accord with the conventions for the D 6h point group symmetry, as pictured
in Fig. 3.10, this site group is based on operators of type ˆ
C ′
2 and ˆ
σ v . The induced
irrep of the six atomic orbitals then becomes
Γ(b 1 C 2v ↑ D 6h ) = A 2u + E 1g + E 2u + B 2g
(4.120)
Since each irrep occurs only once, there is a one-to-one correlation between these
irreps and the cycle index k, which can be retrieved from the D 6h ↓ C 6 subduction
rules :
A 2u −→ A( k = 0)
E 1g −→ E 1 (k =±1)
E 2u −→ E 2 (k =±2)
B 2g −→ B( k = 3)
(4.121)
We will now engage in a more elaborate application of Hückel theory, which
demonstrates the power of this simple model. The purpose is to determine the energy shifts of the eigenvalues when an annulene is brought into a uniform magnetic
field, B. This field is independent of position and time. It can be defined as the “curl”
(or rotation) of a vector potential A, and, in terms of a position vector r from a given
87
Fig. 4.7 Hückel orbital
energy spectrum of benzene
as a function of index k,with
allowed values 0, ±1, ±2, 3
=
1
N
N −1
j =0
α + β
exp(−2πik/N)+ exp(+2πik/N)
= α + 2β cos(2πk/N)
(4.119)
The energies are thus seen to form N discrete levels, which are points on a cosine
curve, as shown in Fig. 4.7. Except for k = 0, and in the case of N even, k = N/2,
all levels E k and E −k are twofold-degenerate. Closed-shell structures thus will be
realized for N = 4n + 2, which is the famous Hückel condition for aromaticity.
These cyclic labels can easily be expanded to the full irrep designations of the D 6h
symmetry group for benzene. The atomic p z -orbitals transform as b 1 in the C 2v site
group. In accord with the conventions for the D 6h point group symmetry, as pictured
in Fig. 3.10, this site group is based on operators of type ˆ
C ′
2 and ˆ
σ v . The induced
irrep of the six atomic orbitals then becomes
Γ(b 1 C 2v ↑ D 6h ) = A 2u + E 1g + E 2u + B 2g
(4.120)
Since each irrep occurs only once, there is a one-to-one correlation between these
irreps and the cycle index k, which can be retrieved from the D 6h ↓ C 6 subduction
rules :
A 2u −→ A( k = 0)
E 1g −→ E 1 (k =±1)
E 2u −→ E 2 (k =±2)
B 2g −→ B( k = 3)
(4.121)
We will now engage in a more elaborate application of Hückel theory, which
demonstrates the power of this simple model. The purpose is to determine the energy shifts of the eigenvalues when an annulene is brought into a uniform magnetic
field, B. This field is independent of position and time. It can be defined as the “curl”
(or rotation) of a vector potential A, and, in terms of a position vector r from a given